FIG. 1 defines three orthogonal coordinate frames, FI, FE, and FN. The first coordinate frame, FI, is called the Inertial Coordinate Frame with axes [XI, YI, ZI] that are fixed with respect to inertial space as represented for example by stellar bodies. The XI and ZI axes of this coordinate frame lie nominally in the equatorial plane of the Earth and the YI axis is nominally coincident with the polar axis of the Earth.
The second coordinate frame, FE, is called the Earth-Fixed Coordinate Frame with axes [XE, YE, ZE]. The XE and ZE axes of this coordinate frame lie nominally in the equatorial plane of the Earth with the ZE axis at the Greenwich meridian and the YE axis nominally coincident with the polar axis of the Earth. The axes XE and ZE, rotate with respect to the Inertial Coordinate Frame, FI, as the earth rotates about its polar axis.
The third coordinate frame, FN, is called the Navigation Coordinate Frame with axes [XN, YN, ZN] where the axes XN and YN are nominally local level at the current position of a vehicle traveling relative to the surface of the Earth with the ZN axis coincident with the local vertical at the current position of the vehicle.
In FIGS. 1 and 2, a “wander-azimuth” angle α is shown which illustrates the rotation of the axes XN and YN in the local level plane relative to the local East and North geodetic axes respectively. Without loss of generality, the local. East, North and Vertical geodetic axes can be assumed as the Navigation Coordinate Frame, FN, in which case the wander azimuth angle α would be identically zero.
In a standard “strapdown” inertial system mechanization, the inertial instruments which are gyroscopes (“gyros”) and accelerometers, are fixed with respect to the vehicle and will have an orientation different than the Navigation Coordinate Frame, FN, defined by the heading of the vehicle with respect to the North axis and the pitch and roll of the vehicle with respect to the local level plane at the current position of the vehicle. In this case the inertial instruments could lie along the axes of an orthogonal Vehicle-Body Coordinate Frame, FB, with axes [XB, YB, ZB] that are rotated with respect to the East, North and Vertical axes by the heading, pitch and roll angles.
In “gimbaled” inertial system mechanizations, the inertial instruments are isolated from the vehicle angular motion by a set of gimbals that use the gyro measurements to realize the stabilization.
One particular gimbaled inertial system mechanization that nominally aligns the axes of the Instrument Coordinate Frame, FA, with axes [XA, YA, ZA], along the East, North and Vertical Navigation Coordinate Frame, FN, axes is called the “Local-Level, North-Slaved” system mechanization. As illustrated in FIG. 3, the Instrument Coordinate Frame, FA, axes are in general slightly misaligned with respect to the Navigation Coordinate Frame, FN, axes due to errors in the measurements of force made by the accelerometers and angular rate made by the gyros along the Instrument Coordinate Frame, FA, axes. These small angular deviations of the Instrument Coordinate. Frame, FA, to the Navigation Coordinate Frame, FN, at the true current position of the vehicle about the East, North and Vertical axes are called tilts and can be denoted respectively by three small angles, [φE, φN, φZ].
In another gimbaled inertial system mechanization, the Instrument Coordinate Frame, FA, is nominally aligned with the axes of the Inertial Coordinate Frame, FI. This particular gimbaled system mechanization is called the “Space Stable” inertial system mechanization. Again small angular errors will exist in achieving alignment of the Instrument Coordinate Frame, FA, with the axes of the Inertial Coordinate Frame, FI, due to errors in the measurements made by the inertial instruments. These errors can be expressed about the axes of the Instrument Coordinate Frame, FA, or about the about the East, North and Vertical navigation axes using the transformation matrix, [InstrTNav], between the Instrument Coordinate Frame, FA, and the Navigation Coordinate Frame, FN.
For the strapdown mechanization referred to above, these same types of angular errors will occur due to inertial instrument measurement errors. For the strapdown system mechanization, the orientation of the Inertial Instrument Frame, FA, can be assumed to be coincident with the Vehicle-Body Coordinate Frame, FB, without loss of generality. In this case the orientation of the Inertial Instrument Frame, FA, is computed with respect to the Navigation Coordinate Frame, FN, using the inertial instrument measurements rather than being rotated to be nominally aligned with the Navigation Coordinate Frame, FN, using the inertial instrument measurements.
In either the strapdown or gimbaled inertial instrument mechanizations, small angular tilt errors [φE, φN, φZ], will exist due to the errors in the measurement made by the gyros and accelerometers. For the strapdown mechanization, the errors will exist in the computed transformation, [NavTInstr]Computed, between the Instrument Coordinate Frame, FA, and the Navigation Coordinate Frame, FN. The relationship between the computed transformation, [NavTInstr]Comp, and the ideal transformation, [NavTInstr]Ideal, can be expressed as: [NavTInstr]Comp=δ[NavTInstr][NavTInstr]Ideal=[I+φ][NavTInstr]Ideal, where:
            [      φ      ]        =          [                                    0                                              φ              U                                                          -                              φ                N                                                                                        -                              φ                U                                                          0                                              φ              E                                                                          φ              N                                                          -                              φ                E                                                          0                              ]        ;            and      ⁢                          [              I        +        φ            ]        =                  [                                            1                                                      φ                U                                                                    -                                  φ                  N                                                                                                        -                                  φ                  U                                                                    1                                                      φ                E                                                                                        φ                N                                                                    -                                  φ                  E                                                                    1                                      ]            .      
For the case of the gimbaled system mechanization, the ideal transformation occurs when the Instrument Coordinate Frame, FA, is rotated so as to be coincident with Navigation Coordinate Frame, FN. In terms of the notation above: [NavTInstr]Ideal=[I] However due to errors in the gyro and accelerometer measurements, the error in realizing the ideal situation for the gimbaled case is expressed by the matrix [I+φ].
Consequently in both the strapdown and gimbaled inertial navigation system mechanization cases, a means of measurement of the orientation error expressed by the three tilt angles about the East, North and Vertical axes [φE, φN, φZ] is beneficial.
A discussion of the nature of the errors for the strapdown and gimbaled inertial navigation system mechanizations discussed above are derived in detail in: “Inertial Navigation System Error-Model Considerations in Kalman Filtering Applications”, by James R. Huddle in Volume 20 of Control and Dynamic Systems edited by C. T. Leondes, Academic Press, 1983, Pp. 293-339, herein incorporated by reference in its entirety. This reference proves the equivalence of error models for the strapdown and gimbaled inertial navigation system error models in so far as the navigation system equations required to implement these system mechanizations are concerned. Consequently, a description in the context of the strapdown inertial navigation system mechanization is readily applied to other inertial navigation systems such that a description for each type of inertial navigation system is not required.
FIG. 4 depicts the signal flow for the strapdown navigation system mechanization. At the left of FIG. 4, measurements of force, [a+g]Instrument, are made by the accelerometers in the Instrument Coordinate Frame, FA, of FIG. 2. The force measurement [a+g]Instrument is the sum of acceleration of the vehicle/navigation system with respect to inertial space plus the force of the gravity vector along the accelerometer sensing axes. To employ these accelerometer measurements for the navigation solution, they must first be transformed to the Navigation Coordinate Frame, FN, via the time varying transformation NavT(t)Instr as shown in the figure. In FIG. 4, the transformed measurements are denoted [a+g]Navigation.
Once the force measurements are expressed in the Navigation Coordinate Frame, FN, then the gravity vector, gNav, in that frame nominally along the local vertical can be subtracted from the measurement to obtain the acceleration, aNav, of the vehicle with respect to inertial space in the Navigation Coordinate Frame, FN.
To realize the function of navigation with respect to the Earth, the vehicle acceleration aNav with respect to inertial space in the Navigation Coordinate Frame, FN, must be processed to obtain the time derivative of vehicle velocity with respect to the Earth taken with respect to the Navigation Coordinate Frame, FN. This conversion is realized through the use of a Coriolis Acceleration Correction as shown in the figure.
Subsequent integration of the corrected measurement produces the vehicle velocity with respect to the Earth, VVehicle/E(t), expressed in the Navigation Coordinate Frame, FN, as shown in FIG. 4.
The computed vehicle velocity with respect to the Earth, VVehicle/E(t), can be transformed via the transformation [EarthT(t)Nav] to the Earth-Fixed Coordinate Frame using knowledge of the current position of the vehicle (latitude Φ, and longitude, λ, as shown in FIG. 1), and integrated to obtain change in position of the vehicle with respect to the Earth in terms of latitude and longitude. Knowledge of the computed latitude of the vehicle with respect to the Earth allows computation of the components (ΩN and ΩZ) of the Earth rotation vector, Ω, with ΩN being the component around the North axis, XN, of the Navigation Coordinate Frame, FN, and ΩZ being the component around the Vertical axis, ZN, of the Navigation Coordinate Frame, FZ.
The computation of the time varying transformation [NavT(t)Instr]Computed, from the Instrument Coordinate Frame, FA, to the Navigation Coordinate Frame, FN, requires knowledge of the angular rate of one frame relative to the other frame. The gyros in the Instrument Coordinate Frame, FA, measure the angular rate of this coordinate frame with respect to inertial space indicated by ωInstrument(t) in FIG. 4. The angular rate of the Navigation Coordinate Frame, FN, with respect to Inertial Space ωNav(t) is the sum of the angular rate of the Earth with respect to Inertial Space, Ω, and the relative angular rate of the Navigation Coordinate Frame, FN, with respect to the Earth. The relative angular rate of the Navigation Coordinate Frame, FN, with respect to the Earth ρNav(t), is computed by dividing the vehicle velocity with respect to the Earth by the local radius of curvature, R, of the Earth at the current position of the vehicle, plus the altitude, h, of the vehicle above the Earth's surface as shown in FIG. 4. The difference of these two angular rate vectors [ωNav-ωInstrument], provides the angular rate of the of the Navigation Coordinate Frame, FN, relative to the Instrument Coordinate Frame, FA, that enables the computation of the required transformation [NavT(t)Instr]Computed shown in FIG. 4.